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# International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

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2191-0294
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Volume 18, Issue 7-8

# A Fractional $q$-difference Equation with Integral Boundary Conditions and Comparison Theorem

Jing Ren
/ Chengbo Zhai
Published Online: 2017-10-28 | DOI: https://doi.org/10.1515/ijnsns-2017-0056

## Abstract

In this article, we mainly prove the existence of extremal solutions for a fractional $q$-difference equation involving Riemann–Lioville type fractional derivative with integral boundary conditions. A comparison theorem under weak conditions is also build, and then we apply the comparison theorem, monotone iterative technique and lower–upper solution method to prove the existence of extremal solutions. Moreover, we can construct two iterative schemes approximating the extremal solutions of the fractional $q$-difference equation with integral boundary conditions. In the last section, a simple example is presented to illustrate the main result.

MSC 2010: 34B18; 33D05

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Published Online: 2017-10-28

Published in Print: 2017-12-20

This paper was supported financially by the Youth Science Foundation of China (11201272), Shanxi Province Science Foundation (2015011005) and 131 Talents Project of Shanxi Province (2015).

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 18, Issue 7-8, Pages 575–583, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339,

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